In a medium bowl, add squash, olive oil, ¼ teaspoon salt, ¼ teaspoon black pepper and stir to combine. Place squash in an even layer on prepared baking sheet.
In a 10-inch cast cast iron skillet, melt butter over medium heat. Add sage leaves and cook until crisp, about 1 minute. Remove sage, chop, and set aside.
Add flour to skillet, whisking to combine. Cook for 1 minute, whisking constantly. Then add milk, nutmeg, remaining 1¾ teaspoon salt, and remaining ¼ teaspoon black pepper, whisking until smooth.
Reduce heat to low. Add 1 cup Asiago cheese and Monterey Jack, stirring until melted. Add fried sage, fresh sage, cooked pasta, and roasted squash, stirring gently to combine.